Category Archives: Mathematics

Introduction to Wasserstein metric (earth mover’s distance)

Table of Contents 1. the cost of moving one particle 2. The cost of moving a finite collection of particles of total mass 1. 3. coupling 4. composing 5. definition of the Wasserstein distance 6. Moving a collection of particles … Continue reading

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Intuitions on problems from Elements of Information Theory Chapter 2

Table of Contents 1. meaning of conditional mutual information 2. comments on some selected problems from Chapter 2 2.1. Problem 2.6 2.2. Problem 2.14 2.3. Problem 2.23 2.4. Problem 2.27 2.5. Problem 2.41 This post is not about sharing solutions … Continue reading

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Doob-Dynkin Lemma for probability space

Table of Contents 1. Proof 1 2. Proof 2 3. Proof 3 4. Uniqueness of H 5. applications Let f, g be measurable functions from a probability space to measurable spaces and respectively. Consider the following three conditions: (1) (in … Continue reading

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Proofs of Doob-Dynkin Lemma

Table of Contents 1. uniqueness of H 2. counterexamples 3. proof when g has a countable range, with nice Y 4. proof when Y is the Cantor space 5. proof when Y is the unit interval 6. proof for standard … Continue reading

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Small introduction to Hausdorff distance

Table of Contents 1. motivation and intuition 2. coming up with the two definitions of the Hausdorff distance 3. more exercises Update: I realized that parts of this post focus on unimportant small details. Sorry for that, you may skip … Continue reading

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mod 0 equivalence of sub-sigma-algebras and gotchas

Table of Contents 1. closure of a sub-sigma-algebra 2. closed sub sigma algebra 3. mod zero equivalence of sub sigma algebras When we are building some theory involving measures, there are cases where we should ignore null sets (sets of … Continue reading

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using org-mode export feature to publish to wordpress with math equations

Table of Contents some preliminary search and replace some idea on how to write a math blog post using org-mode, AUCTeX, latex-preview If you are using the latest official version of Emacs and the latest version of org-mode, which is … Continue reading

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Shannon’s entropy of random variables and partitions

Table of Contents 1. entropy for dummies 1.1. definition of entropy of r.v. and joint entropy 1.2. the town 1.3. intuition for entropy and its properties 1.4. intuition for the notion of surprisal 2. intuition for conditional entropy and its … Continue reading

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How to use Unicode in LaTeX (by LuaTeX or XeTeX)

Table of Contents 1. before we begin 2. LuaLaTeX vs XeLaTeX and what are they? 3. first example 4. how to experiment with adjusting font features 5. in summary 6. how to change names 7. some differences with LuaLaTeX The … Continue reading

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difference between “a set of” and “the set of”

This may be obvious to some people, but not to some students who signed up for a math lecture in English, in a country that doesn’t speak English. The set of positive integers is the set of all positive integers. … Continue reading

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